Why manifolds?

Motivation We know well how to compute with planar domains and functions many graphics and geometric modeling applications involve domains of nontrivial topology closed surfaces, configuration spaces, light fields … Manifolds: a tool for constructing algorithms do computations on planar domains then blend together; how to blend smoothly? Manifolds: a tool for understanding algorithms why do we see (or do not see) problems when computing with complex domains?

Domains Geometric modeling construct smooth surfaces Can get unique combinations of properties understand how to build smooth global parametrizations Animation smoothly interpolate motions represent config. spaces for motion editing Rendering assemble smooth lightfields from different views, represent BRDFs

Constructing smooth surfaces Can get unique combinations of properties: arbitrary smoothness, local support, flexibility; compare: even C2 subdivision is very difficult; add local charts anywhere you want

Parametrization Global parametrization Gives us tools to get smoothness everywhere Gu and Yau, 2003Ray, Li, Levy, Sheffer, Alliez, 2005

Parametrization Essential question: what is a smooth function on a mesh?

Parametrization Why this one global algorithm works better than another? parametrization derivative approximations Khodakovsky and Schröder

Examples Animation Configuration spaces are manifolds Rendering Light fields are manifolds Surface modeling and parameterization

Goals for surface modeling A high-order surface construction Important for geometric and numerical computation Desirable features or smoothness At least 3-flexibility at vertices Closed-form smooth local parameterizations Can handle arbitrary control meshes Good visual quality Easy to implement

Smoothness smoothness A standard goal in CAGD important for high-accuracy computation Computing surface properties : needed for normal : needed for curvatures, reflection lines; : needed for curvature variation;

Flexibility Ability to represent local geometry Property of basis function, instead of the surface Two-Flexibility: any desired curvature at any point 1-flexible2-flexible Todo: replace with shaded picture

Local Parameterization Explicit smooth local parameterization For any point, there is an explicit formula defining the surface in a neighborhood of this point Simplifies many tasks Defining functions on surfaces Integration over surfaces Surface-surface intersections Computing geodesics

Spline-based Approach Construct surface patch for each face Find smooth local parameterization for every point Difficult to guarantee smoothness for points on patch boundaries

Manifold-based Approach Construct overlapping charts covering the mesh Build local geometry approximating the mesh on each chart Find blending function for each chart Get the surface by blending local geometry ……

Previous Work High-order spline patches S-patches [Loop and DeRose 1989] DMS splines [Seidel 1994] Freeform splines [Prautzsch 1997] TURBS [Reif 1998] C2 flexible subdivision surfaces G2 subdivision [Prautzsch and Umlauf 1996] Manifold-based approach [Grimm and Hughes 1995] [Navau and Garcia 2000] [Grimm 2002] [Ying, Zorin 2004] [Qin, Gu, He, 2005] [Grimm 2005]